Theorem apsym 8749

Description: Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)

Assertion

Ref	Expression
apsym	|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )

Proof of Theorem apsym

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 8138	. . 3 |- ( B e. CC -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
2	1	adantl 277	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
3		cnre 8138	. . . . . 6 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
4	3	ad3antrrr 492	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
5		simplrl 535	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> x e. RR )
6		simplrl 535	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> z e. RR )
7	6	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> z e. RR )
8		reaplt 8731	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( x # z <-> ( x < z \/ z < x ) ) )
9	5, 7, 8	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x # z <-> ( x < z \/ z < x ) ) )
10		reaplt 8731	. . . . . . . . . . . . 13 |- ( ( z e. RR /\ x e. RR ) -> ( z # x <-> ( z < x \/ x < z ) ) )
11	7, 5, 10	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( z # x <-> ( z < x \/ x < z ) ) )
12		orcom 733	. . . . . . . . . . . 12 |- ( ( x < z \/ z < x ) <-> ( z < x \/ x < z ) )
13	11, 12	bitr4di 198	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( z # x <-> ( x < z \/ z < x ) ) )
14	9, 13	bitr4d 191	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x # z <-> z # x ) )
15		simplrr 536	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> y e. RR )
16		simplrr 536	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> w e. RR )
17	16	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> w e. RR )
18		reaplt 8731	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( y # w <-> ( y < w \/ w < y ) ) )
19	15, 17, 18	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y # w <-> ( y < w \/ w < y ) ) )
20		reaplt 8731	. . . . . . . . . . . . 13 |- ( ( w e. RR /\ y e. RR ) -> ( w # y <-> ( w < y \/ y < w ) ) )
21	17, 15, 20	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( w # y <-> ( w < y \/ y < w ) ) )
22		orcom 733	. . . . . . . . . . . 12 |- ( ( y < w \/ w < y ) <-> ( w < y \/ y < w ) )
23	21, 22	bitr4di 198	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( w # y <-> ( y < w \/ w < y ) ) )
24	19, 23	bitr4d 191	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y # w <-> w # y ) )
25	14, 24	orbi12d 798	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x # z \/ y # w ) <-> ( z # x \/ w # y ) ) )
26		apreim 8746	. . . . . . . . . 10 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
27	5, 15, 7, 17, 26	syl22anc 1272	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
28		apreim 8746	. . . . . . . . . 10 |- ( ( ( z e. RR /\ w e. RR ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) <-> ( z # x \/ w # y ) ) )
29	7, 17, 5, 15, 28	syl22anc 1272	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) <-> ( z # x \/ w # y ) ) )
30	25, 27, 29	3bitr4d 220	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) ) )
31		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> A = ( x + ( _i x. y ) ) )
32		simpllr 534	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> B = ( z + ( _i x. w ) ) )
33	31, 32	breq12d 4095	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) ) )
34	32, 31	breq12d 4095	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( B # A <-> ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) ) )
35	30, 33, 34	3bitr4d 220	. . . . . . 7 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> B # A ) )
36	35	ex 115	. . . . . 6 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( A = ( x + ( _i x. y ) ) -> ( A # B <-> B # A ) ) )
37	36	rexlimdvva 2656	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( A # B <-> B # A ) ) )
38	4, 37	mpd 13	. . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( A # B <-> B # A ) )
39	38	ex 115	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) -> ( B = ( z + ( _i x. w ) ) -> ( A # B <-> B # A ) ) )
40	39	rexlimdvva 2656	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) -> ( A # B <-> B # A ) ) )
41	2, 40	mpd 13	1 |- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082  (class class class)co 6000   CCcc 7993   RRcr 7994   _ici 7997   + caddc 7998   x. cmul 8000   < clt 8177   # cap 8724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725

This theorem is referenced by:  addext  8753  mulext  8757  ltapii  8778  ltapd  8781  aptap  8793  apdivmuld  8956  div2subap  8980  recgt0  8993  prodgt0  8995  irrmulap  9839  pwm1geoserap1  12014  absgtap  12016  geolim  12017  geolim2  12018  geo2sum2  12021  geoisum1c  12026  tanaddap  12245  egt2lt3  12286  sqrt2irraplemnn  12696  1sgm2ppw  15663  triap  16356  apdiff  16375